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QM Model

Concept of Atomic Orbital

When Heisenberg put forward his Uncertainty principle, which said that, at any one time, it is impossible to calculate both the momentum and the location of an electron in an atom; it is only possible to calculate the probability of finding an electron within a given space.

And thus the Quantum Mechanical Model redefined the way electrons travel, according to this approach, we cannot simply say that the electron exists at a particular point in space. Instead of defining a particular path, it proposed some region in space around the nucleus, called an orbital, where the probability of finding an atom is maximum. Thus the electron doesn't always remain at a definite distance from the nucleus.

The Energy levels classify the Orbitals based of their proximity towards the nucleus, these are represented as \[\text{K }(n=1), \text{L }(n=2), \text{M }(n=3), \text{N }(n=4), \text{O }(n=5), \cdots\] The lowest energy level is K or 1, the next being L or 2 and so on. Thus for an electron the energy level describes the path of the electron and the energy of the electron given by the equation;

Where \(h\) is the plank's constant, \(c\) is the speed of light, \(R_\infty\) is the Rydberg's constant, \(Z\) is the atomic number of the element and the electron is present in the \(n^{th}\) energy level.

The energy level are sub-divided into sub-shells, which are designated as \(s\), \(p\), \(d\) and \(f\), and the number of sub-shells in each energy level is given by the number itself, for example: the K-energy level has only \(1\) sub-shell \(s\), and the L-energy level has \(2\) sub-shells \(s\), and \(p\).

The sub-shells are precisely defined with the help of quantum numbers, which govern the number of orbitals in each sub-shell. The Magnetic Quantum Number\((m_l)\) visualizes the behavior of an electron under the influence of a magnetic field(like earth). We know that the movement of electric charge can generate a Magnetic field, and under the influence of an external magnetic field the electrons tend to orient themselves in certain regions around the nucleus(called orbitals), which is why this quantum number gives the number of orbitals in a particular sub-shell.

The values of the Magnetic Quantum Number depends upon the Azimuthal Quantum Number\((l)\), for example if the Azimuthal Quantum Number of an atom is \(l\), then the Magnetic Quantum Numbers range from:

\[m_l= -l,( -l+1), . . . , 0 , . . . (l-1), l\]

So, there are \(2l+1\) values of \(m_l\) for a given value of \(l\), i.e. there will be \(2l+1\) orbitals. According to the Magnetic Quantum Number, the number orbitals in: